# Let there be the function #f(alpha)= (L( v_a - v_0 sin alpha))/(v_0 cos alpha)# Where #L# ,#v_a# and #v_o# are constants. Determine #alpha# such that #f(alpha)# is minimal ?

##### 2 Answers

#F(alpha) = f(alpha)/L qquad F'(alpha) = (f'(alpha))/L qquad " etc "#

To optimise

# F_alpha = v_a/v_o secalphatanalpha - sec^2alpha = 0#

Critical points can, therefore, only occur under these rules:

# sinalpha = v_o/v_a qquad qquad implies {(v_o lt= v_a),(cos alpha = sqrt(v_a^2 - v_o^2)/v_a),(tanalpha = v_o/sqrt(v_a^2 - v_o^2) ):}#

To determine the max or min nature of the Critical Point, grab the second derivative of

With assumption

Accordingly, there is a min when this occurs:

# sinalpha = v_o/v_a #

This min will be periodic.

I have solved this way. See the answer below: